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[[125,6,13]] d ≤
n
125
k
6
d
13
kd²/n
8.112
w
6
X/Z
1.08
g
0.007
r
5.831
layers
2
swaps
577

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Distance

X/Z asymmetry 1.08 · d_X ≤ 14, d_Z ≤ 13 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 14 qubits)
[0, 7, 41, 43, 46, 49, 56, 58, 66, 73, 87, 97, 111, 123]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 13 qubits)
[0, 2, 4, 17, 25, 26, 30, 39, 50, 61, 83, 97, 122]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 5–6 (mean 5.967)
qubit degrees H_X 1–3 (mean 2.832) · H_Z 1–3 (mean 2.864)
trapping sets H_X (1,1)×1 (2,1)×1 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 1 (1,2): 19 (1,3): 105 (2,1): 1 (2,2): 11 (2,3): 175 (2,4): 698 (3,1): 1 (3,2): 36 (3,3): 322 (3,4): 1900 (3,5): 5809 (3,6): 323 (3,7): 821
trapping sets H_Z (1,1)×2 (2,2)×12 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 2 (1,2): 13 (1,3): 110 (2,2): 12 (2,3): 126 (2,4): 752 (3,1): 1 (3,2): 28 (3,3): 294 (3,4): 1410 (3,5): 6444 (3,6): 244 (3,7): 908
witness diameter X 8.0623 · Z 8.0623 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @vprusso · Inherited from the laid-out parent codes/126-6-14.json: the parent's coordinates with the entry of the removed qubit (parent index 55) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 5.831
X checkZ checkqubit site (64)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 577 nearest-neighbor SWAPs per round in total, at most 9 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus E-PEEL transform of codes/126-6-14.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
family other (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[125,6,13]] — priced calculus E-PEEL transform of codes/126-6-14.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/126-6-14.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/126-6-14.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/126-6-14.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/126-6-14.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 59 (max weight 6) · Z-checks 60 (max weight 6)
H_X (59 checks, sparse supports)
[1, 24, 48, 63, 96, 124] [0, 2, 59, 64, 76, 107] [3, 36, 51, 65, 108, 110] [4, 30, 53, 66, 102, 104] [1, 5, 33, 67, 82, 105] [6, 34, 55, 68, 106, 115] [2, 7, 61, 69, 85, 115] [3, 8, 45, 63, 70, 116] [9, 46, 57, 71, 117, 118] [4, 10, 39, 62, 72, 111] [11, 40, 58, 73, 112, 113] [5, 12, 42, 74, 91, 113] [6, 13, 43, 66, 75, 114] [14, 44, 60, 76, 86, 121] [7, 15, 20, 66, 77, 94] [8, 16, 53, 67, 78, 121] [9, 17, 54, 68, 79, 122] [10, 18, 48, 64, 80, 118] [11, 19, 49, 65, 81, 119] [20, 50, 61, 82, 92, 120] [0, 12, 21, 65, 83, 100] [13, 22, 51, 72, 84, 120] [14, 23, 52, 73, 85, 95] [15, 24, 29, 72, 86, 103] [4, 16, 25, 73, 74, 87] [17, 26, 58, 75, 88, 124] [1, 18, 27, 68, 69, 89] [19, 28, 55, 70, 90, 123] [20, 29, 56, 71, 91, 101] [2, 21, 30, 70, 92, 109] [3, 22, 31, 71, 80, 93] [23, 32, 57, 81, 94, 104] [24, 33, 38, 80, 95, 112] [10, 25, 34, 81, 83, 96] [11, 26, 35, 82, 84, 97] [5, 27, 36, 75, 77, 98] [6, 28, 37, 76, 78, 99] [29, 38, 60, 79, 100, 110] [7, 30, 39, 78, 101, 117] [8, 31, 40, 79, 89, 102] [9, 32, 41, 62, 90, 103] [33, 42, 47, 89, 104, 119] [18, 34, 43, 90, 92, 105] [19, 35, 44, 91, 93, 106] [12, 36, 45, 84, 86, 107] [13, 37, 46, 85, 87, 108] [14, 38, 47, 63, 88, 109] [15, 39, 48, 87, 110, 122] [16, 40, 49, 88, 98, 111] [17, 41, 50, 64, 99, 112] [27, 43, 51, 99, 101, 113] [28, 44, 52, 100, 102, 114] [21, 45, 53, 93, 95, 115] [22, 46, 54, 94, 96, 116] [25, 49, 55, 97, 107, 118] [26, 50, 56, 69, 108, 119] [37, 52, 57, 109, 111, 120] [31, 54, 58, 103, 105, 121] [35, 56, 60, 77, 116, 123]
H_Z (60 checks, sparse supports)
[0, 10, 41, 62, 64, 83] [1, 8, 47, 63, 67, 89] [2, 18, 50, 64, 69, 92] [3, 19, 21, 65, 70, 93] [4, 13, 15, 66, 72, 87] [5, 16, 67, 74, 98] [6, 17, 27, 68, 75, 99] [7, 27, 56, 69, 77, 101] [8, 28, 30, 70, 78, 102] [9, 29, 31, 71, 79, 103] [10, 22, 24, 72, 80, 96] [11, 23, 25, 73, 81, 97] [12, 25, 59, 74, 83, 107] [13, 26, 36, 75, 84, 108] [2, 14, 37, 76, 85, 109] [15, 36, 60, 77, 86, 110] [16, 37, 39, 78, 87, 111] [17, 38, 40, 79, 88, 112] [18, 31, 33, 80, 89, 105] [19, 32, 34, 81, 90, 106] [5, 20, 35, 77, 82, 91] [21, 34, 61, 83, 92, 115] [22, 35, 45, 84, 93, 116] [7, 23, 46, 85, 94, 117] [14, 24, 45, 63, 86, 95] [25, 46, 48, 87, 96, 118] [26, 47, 49, 88, 97, 119] [27, 40, 42, 89, 98, 113] [28, 41, 43, 90, 99, 114] [12, 29, 44, 86, 91, 100] [20, 30, 43, 66, 92, 101] [31, 44, 53, 93, 102, 121] [15, 32, 54, 94, 103, 122] [23, 33, 53, 67, 95, 104] [1, 34, 54, 68, 96, 105] [35, 55, 97, 106, 123] [0, 36, 49, 65, 98, 107] [37, 50, 51, 99, 108, 120] [21, 38, 52, 95, 100, 109] [29, 39, 51, 72, 101, 110] [4, 40, 52, 73, 102, 111] [24, 41, 58, 103, 112, 124] [4, 32, 42, 62, 74, 104] [5, 43, 58, 75, 105, 113] [6, 44, 59, 76, 106, 114] [2, 45, 55, 70, 107, 115] [3, 46, 56, 71, 108, 116] [30, 47, 57, 104, 109, 117] [3, 38, 48, 63, 80, 110] [10, 49, 57, 81, 111, 118] [11, 33, 50, 82, 112, 119] [11, 12, 51, 65, 84, 113] [13, 52, 61, 85, 114, 120] [6, 7, 53, 66, 78, 115] [8, 54, 60, 79, 116, 121] [9, 18, 55, 68, 90, 118] [19, 42, 56, 91, 119, 123] [20, 22, 57, 71, 94, 120] [14, 16, 58, 73, 88, 121] [0, 28, 60, 76, 100, 123]
Code ID 125-6-13 · download JSON · raw on GitHub